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A Stereo Image

A Stereo Image. The space of all stereo images. The space/geometry of all stereo/epipolar images/cameras. S. Seitz, J. Kim, The Space of All Stereo Images , IJCV 2002 / ICCV 2001. T. Pajdla, Epipolar Geometry of Some Non-Classical Cameras , Slovenian

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A Stereo Image

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  1. A Stereo Image

  2. The spaceof all stereo images The space/geometry of all stereo/epipolar images/cameras • S. Seitz, J. Kim,The Space of All Stereo Images, IJCV 2002 / ICCV 2001. • T. Pajdla, Epipolar Geometry of Some Non-Classical Cameras, Slovenian • Computer Vision Winter Workshop, 2001. or The space (geometry) of ray sets (cameras) that allow row-based stereo analysis

  3. Outline 2) Generalized / non-classical camera 1) Regular stereo 3) Stereo with generalized cameras (an example) 4) When can we compute stereo from generalized cameras?

  4. A Stereo Image

  5. Stereoscopic illusion Real world Stereo Illusions

  6. Stereoscopic illusion

  7. Stereoscopic Imaging • Key property: horizontal parallax • Enables stereoscopic viewing and stereo algorithms Thank you, Steve Seitz

  8. Capturing stereo pairs Left camera Right camera

  9. 2) Color glasses p 0 3) Polarized glasses • Temporally synchronized • screen and glasses Display different image for each eye 1) Separating with vertical paper

  10. Parallax and disparity Aligned images Original images L 1 row R Z=1 Z=2 Z=0 Z=0 Z=1 Z=2 L L Z R R Disparity (1D) Alignment  Measure depth with respect to this plane Parallax (3D)

  11. Displaying disparity

  12. Stereo Algorithms Photogrammetry (generating maps)

  13. Stereoscopic Imaging • Key property: horizontal parallax • Enables stereoscopic viewing and stereo algorithms Thank you, Steve Seitz

  14. RectificationorWhy we can focus on 1-row Line corresponds to line (“epipolar lines”) Rectified images: epipolar lines are image rows Homography ? ?

  15. Rectification (cont.)

  16. Outline 2) Generalized / non-classical camera 1) Regular stereo 3) Stereo with generalized cameras (an example) 4) When can we compute stereo from generalized cameras?

  17. Geometric camera model We model camera as Mapping: world points  image points (pixels) • We model only the “projection” operation • We do not model: light, color, lens blurring, etc.

  18. Important property: every ray from the ray set of the camera projects to one point Example: pinhole camera model(usual camera) This projection operation is commonly described by a 3x4 projective matrix. For our purposes the following is more convenient: • A pinhole camera is defined by: • set of rays (starting from the camera center) • mapping from this rays to the image plane

  19. A generalized camera maps rays to image points For our purposes camera  set of rays Generalization of the camera model Classical camera • One center of projection • Image surface (film) is planar Generalized (ray-projective) camera • Multiple centers of projection (origins of rays) • Image surface is arbitrary

  20. Xerox machine(non-classical cameras, example 1) As a multi-prospective camera:

  21. Imax 3D Incredibly realistic three-dimensional images are projected onto the giant IMAX screen with such realism that you can hear the audience gasp as they reach out to grab the almost touchable images. Imax Dome Experience wraps the audience in images of unsurpassed size and impact, providing an amazing sense of involvement. thanks to Shmuel Peleg, Hebrew University of Jerusalem Existing Immersive Technology: Imax®Can Not Combine Full FOV and Stereopsis

  22. Pushbroom camera(non-classical cameras, example 2) • 1D projective sensor • … translating Advantage: large field of view in one dimension

  23. y x Pushbroom camera(non-classical cameras, example 2) The generalized camera model: X direction - parallel projectionY direction - perspective projection NotionGenerator – the set of all ray origins For other cameras, generator can be a 2D surface

  24. thanks to Shmuel Peleg, Hebrew University of Jerusalem Pushbroom camera(non-classical cameras, example 2) Imaging process Images from usual camera t t+1 t+2 Image from generalized camera Virtual generalized camera = device (usual camera) + software

  25. t Non-classical cameras • Implementation through cuts of 3D video arrays • Take images while moving a usual camera • Stack them into 3D array • Take a cut along the “time” dimension

  26. thanks to Shmuel Peleg, Hebrew University of Jerusalem Circular projective camera(non-classical cameras, example 3) Move “1D sensor” along a circlerecord on a cylinder Advantage: complete 360° horizontal view

  27. Circular projective camera(non-classical cameras, example 3) The generalized camera model: Note: Generator is a circle

  28. Outline 2) Generalized / non-classical camera 1) Regular stereo 3) Stereo with generalized cameras (an example) 4) When can we compute stereo from generalized cameras?

  29. thanks to Steve Seitz, University of Washington Generalized stereo: an example Inward-facing camera, moving around an object

  30. Output: 2 symmetric cuts of 3D video array thanks to Steve Seitz, University of Washington Images for both eyes • Input: video sequence

  31. thanks to Steve Seitz, University of Washington Results: red-blue stereo image

  32. thanks to Steve Seitz, University of Washington Results: 3D reconstruction Using usual algorithm (built for usual camera)with non-classical images

  33. Ray geometry How does the set of rays look? Pixel = ray There is a “blind” area in the center of the scene

  34. Ray geometry What rays go through a point in the scene?How disparity depends on depth?

  35. Outline 2) Generalized / non-classical camera 1) Regular stereo 3) Stereo with generalized cameras (an example) 4) When can we compute stereo from generalized cameras?

  36. Choose “ground” plane Z=0 L R PARRALAX / DISPARITY L R Z=0 Z=1 Z=2 Parallax and disparity in Cyclographs(review)

  37. D The generalized camera model • D - image surface; P – ray space • A view (camera) is a function V: DP • Does not include: • Multiple rays to 1-image point • Curved light paths (mirror, lens)

  38. A row (row-continuity) A row is the set of points in one view image that corresponds to a ray of the other view. For rectified images: rows are image rows Infinitesimality: • Row has width  0 • Row has no holes

  39. D1 D2 Rays V(u1,v1) and V(u2,v2) intersect v1=v2 Stereo: basic constraints (u2,v2) (u1,v1) row v =v2 row v =v1

  40. Basic stereo constraints + row-continuity Surfaces V(*,v1) and V(*,v2) , where v1, and v2 are corresponding rows, intersect in a surface(not a curve) D1 D2 row v =v1 (u2,v2) (u1,v1) row v =v2

  41. Example: The intersection of epipolar planes The red plane ”intersects” the blue plane

  42. Ruled surfaces:

  43. Ruled surfaces: examples • Generalized cylinder • Generalized cone

  44. The most important slide Doubly ruled surface • Left camera “ruling” the scene • Right camera “ruling” the scene

  45. A hyperboloid Doubly Ruled surfaces: examples A plane

  46. hyperboloid hyperbolic paraboloid plane Theorem (D. Hilbert ):The only doubly ruled surfaces are:

  47. The Doubly Ruled Surfacesof cyclograph

  48. SUMMARY The space (geometry) of ray sets (cameras) that allow row-based stereo analysis are doubly ruled surfaces

  49. Stereo Image Spiral mirror acquiring right eye panorama Spiral mirror acquiring left eye panorama Optical center viewing circle

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